Fractal Weyl law for Linux Kernel Architecture

TL;DR

This paper investigates the spectral properties of the Google matrix derived from Linux Kernel procedure call networks, revealing a fractal Weyl law consistent with quantum chaotic systems and indicating localization of eigenmodes.

Contribution

It demonstrates the applicability of the fractal Weyl law to directed networks like the Linux Kernel, linking spectral properties to network fractal dimensions.

Findings

Spectrum follows fractal Weyl law with exponent ~0.63

Eigenmodes are localized on principal nodes

Fractal Weyl law likely applies to other directed networks with fractal dimension less than 2

Abstract

We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be $\nu \approx 0.63$ that corresponds to the fractal dimension of the network $d \approx 1.2$. The eigenmodes of the Google matrix of Linux Kernel are localized on certain principal nodes. We argue that the fractal Weyl law should be generic for directed networks with the fractal dimension $d<2$.